4. Contact ratio: why gears are quiet or loud
The transverse contact ratio εα is the average number of tooth pairs carrying load at any instant. It is a length divided by a pitch: how far along the line of action contact persists, over how far apart the teeth are.
It has to be above 1, or at some instant no tooth is in mesh at all and the drive stops being a drive. In practice you want 1.4 and above.
| εα | What it means |
|---|---|
| < 1.0 | Not a gear pair. The mesh breaks. |
| 1.0 – 1.2 | Legal but rough. One tooth carries everything for most of the cycle. |
| 1.4 – 1.8 | The normal range for a standard spur pair. |
| > 1.8 | Quiet and smooth — many teeth, or a high-contact-ratio design. |
Why it decides noise
Between one and two, the number of teeth in mesh alternates: one pair, then two, then one again. The mesh stiffness therefore alternates too, and a stiffness that changes at tooth-passing frequency is precisely a source of vibration. Push the ratio above 2 and the load hand-over becomes smooth, which is why high-contact-ratio gearing is used where noise matters more than efficiency.
It also feeds the flank rating directly, through the contact ratio factor:
A fine-pitch pair with many teeth — watch the contact ratio climb well past 1.8 while the tooth gets small enough that the root becomes the worry instead.